diff options
Diffstat (limited to 'theories/Numbers/Integer/BigZ/ZMake.v')
-rw-r--r-- | theories/Numbers/Integer/BigZ/ZMake.v | 62 |
1 files changed, 50 insertions, 12 deletions
diff --git a/theories/Numbers/Integer/BigZ/ZMake.v b/theories/Numbers/Integer/BigZ/ZMake.v index 6d14337ab..fb760bdf5 100644 --- a/theories/Numbers/Integer/BigZ/ZMake.v +++ b/theories/Numbers/Integer/BigZ/ZMake.v @@ -271,23 +271,23 @@ Module Make (N:NType) <: ZType. unfold square, to_Z; intros [x | x]; rewrite N.spec_square; ring. Qed. - Definition power_pos x p := + Definition pow_pos x p := match x with - | Pos nx => Pos (N.power_pos nx p) + | Pos nx => Pos (N.pow_pos nx p) | Neg nx => match p with | xH => x - | xO _ => Pos (N.power_pos nx p) - | xI _ => Neg (N.power_pos nx p) + | xO _ => Pos (N.pow_pos nx p) + | xI _ => Neg (N.pow_pos nx p) end end. - Theorem spec_power_pos: forall x n, to_Z (power_pos x n) = to_Z x ^ Zpos n. + Theorem spec_pow_pos: forall x n, to_Z (pow_pos x n) = to_Z x ^ Zpos n. Proof. assert (F0: forall x, (-x)^2 = x^2). intros x; rewrite Zpower_2; ring. - unfold power_pos, to_Z; intros [x | x] [p | p |]; - try rewrite N.spec_power_pos; try ring. + unfold pow_pos, to_Z; intros [x | x] [p | p |]; + try rewrite N.spec_pow_pos; try ring. assert (F: 0 <= 2 * Zpos p). assert (0 <= Zpos p); auto with zarith. rewrite Zpos_xI; repeat rewrite Zpower_exp; auto with zarith. @@ -300,18 +300,32 @@ Module Make (N:NType) <: ZType. rewrite F0; ring. Qed. - Definition power x n := + Definition pow_N x n := match n with | N0 => one - | Npos p => power_pos x p + | Npos p => pow_pos x p end. - Theorem spec_power: forall x n, to_Z (power x n) = to_Z x ^ Z_of_N n. + Theorem spec_pow_N: forall x n, to_Z (pow_N x n) = to_Z x ^ Z_of_N n. Proof. - destruct n; simpl. rewrite N.spec_1; reflexivity. - apply spec_power_pos. + destruct n; simpl. apply N.spec_1. + apply spec_pow_pos. Qed. + Definition pow x y := + match to_Z y with + | Z0 => one + | Zpos p => pow_pos x p + | Zneg p => zero + end. + + Theorem spec_pow: forall x y, to_Z (pow x y) = to_Z x ^ to_Z y. + Proof. + intros. unfold pow. destruct (to_Z y); simpl. + apply N.spec_1. + apply spec_pow_pos. + apply N.spec_0. + Qed. Definition sqrt x := match x with @@ -451,4 +465,28 @@ Module Make (N:NType) <: ZType. rewrite spec_0, spec_m1. symmetry. rewrite Zsgn_neg; auto with zarith. Qed. + Definition even z := + match z with + | Pos n => N.even n + | Neg n => N.even n + end. + + Definition odd z := + match z with + | Pos n => N.odd n + | Neg n => N.odd n + end. + + Lemma spec_even : forall z, even z = Zeven_bool (to_Z z). + Proof. + intros [n|n]; simpl; rewrite N.spec_even; trivial. + destruct (N.to_Z n) as [|p|p]; now try destruct p. + Qed. + + Lemma spec_odd : forall z, odd z = Zodd_bool (to_Z z). + Proof. + intros [n|n]; simpl; rewrite N.spec_odd; trivial. + destruct (N.to_Z n) as [|p|p]; now try destruct p. + Qed. + End Make. |